Simplify the following expression and state the condition under which the simplification is valid. $k = \dfrac{a^2 - 49}{a - 7}$
Solution: First factor the polynomial in the numerator. The numerator is in the form ${a^2} - {b^2}$ , which is a difference of two squares so we can factor it as $({a} + {b})({a} - {b})$ $ a = a$ $ b = \sqrt{49} = -7$ So we can rewrite the expression as: $k = \dfrac{({a} {-7})({a} + {7})} {a - 7} $ We can divide the numerator and denominator by $(a - 7)$ on condition that $a \neq 7$ Therefore $k = a + 7; a \neq 7$